Counterbores and chamfering tools are well-known for frequently causing problems when used in machining parts held in a rotating chuck. If the counterbore or chamfering tool is held, for example, in the tail stock of a lathe and the part to be shaped is in the head stock, it is not uncommon for the centerline of the part to not precisely align with the centerline of the collet holding the part. When this happens, the chamfering tool will tend to shape only one portion of the aperture in the part to be shaped leaving the remainder intact. The tool can also provide an egg-shaped chamfer depending on the degree that the centerline of the part is off the centerline of the collet holding the part. In the precision machining or chamfering of small parts, it is extremely difficult to properly align the chamfering tool with the aperture in the part which is to be shaped.